In the following, we adapt to our framework some classic results of viscosity solutions theory that can be found, e.g., in [1]}, [2]}. In particular, the hybrid framework that we propose is related to hybrid control [3]} and somehow to
the mathematical switching hysteresis models [4]}. The need of memory feature, associated with the optimal visiting and dynamic programming and Hamilton-Jacobi equations has been presented in [5]}, where a continuous hysteresis memory was introduced. The use of a switching/discontinuous/hybrid memory, as in the present paper, was instead used for a one-dimensional optimal visiting problem on a network in [6]}. For switching hybrid control problems, as well as for differential games, related to the model here presented, and in connection with Hamilton-Jacobi equations, we refer to [7]} and to [8]}. A more general discussion is done in [9]} (similar formulations for the deterministic case have also been proposed in [3]}, [11]}).
